Theory Mathematical analysis of intermittent dual countercurrent chromatography has been proposed as follows: 1) General Model Retention time of the solute peak for intermittent dual CCC is mathematically analyzed. The method elutes each phase alternately through the opposite ends of the separation column at a given interval and flow rate while the sample solution is charged at the middle portion of the column. At the hydrodynamic equilibrium state established in each elution mode, the separation channel with a cross sectional area Ac is divided into the area occupied by the lower phase, AL, and the upper phase, AU. Here we assume that the two phases are uniformly distributed at a given volume ratio throughout the channel. The lower phase is introduced from the left terminal at a flow rate of FL ml/min and the upper phase from the right terminal at a flow rate of FU ml/min intermittently at given intervals, ti, L and ti, U, respectively. Then, the analysis of the solute peak motion in the channel is divided into two parts, i.e., for the forward (from left to right) elution and the backward (right to left) elution as follows: For lower phase elution, the lower mobile phase is introduced from the left side of the channel at FLml/min flows through the channel at a linear velocity of uL cm/min, uL= (FL/AC)(AU,L+AL,L)/AL,L = (FL/AC)(BL+1) (1) where BL = AU,L/AL,L (volume ratio of two phases when lower phase mobile). Then the solute in the lower phase moves forward through the channel at a rate of uX,L cm/min according to its partition coefficient, K, and the volume ratio of the two phases in the channel, uX,L = (FL/AC)(BL+1)/(1+K BL) (2) where K = concentration of solute in the upper phase divided by that in the lower phase. Similarly in the backward elution, the upper mobile phase introduced from the right end of the channel at FU ml/min flows through the channel at a linear velocity of uU cm/min, uU = (FU/AC)(AU,U+AL,U)/AU,U = (FU/AC)(1+1/BU) (3) where BU = AU,U/AL,U (volume ratio of two phases when upper phase mobile) and the solute in the mobile upper phase moves backward through the channel at a rate of uX,U cm/min according to K (concentration of solute in the upper phase divided by that in the lower phase) and the two-phase volume ratio in the channel, or uX,U = (FU/AC)(1+1/BU)KBU/(1+KBU) = (FU/AC)K(1+BU)/(1+KBU) (4). Then, from Eq (2) and (4) the average linear velocity (uX,i) of the solute peak is given by uX,i = (uX,L ti,L-uX,U ti,U)/(ti,L + ti,U) = (FL/A) ti,L (BL+1)/(1+KBL) - (FU/A) ti,U K(1+BU)/(1+KBU) /(ti,L + ti,U) (5) where ti, L and ti, U indicate the unit programmed time for forward and backward elution, respectively. Therefore, the retention time (tR) of solute at the end of the unit programmed dual elution is computed from the following equation, tR = L/uX,i =0.5 Vc(ti,L + ti,U)/FL ti,L (1+BL)/(1+KBL) - FU ti,U K(1+BU)/(1+KBU) (6) where L is a half length of the channel and Vc, the total capacity of the channel where Vc = 2LAc. When uX,L>uX,U or uX,i>0, the solute peak moves forward and is eluted from the right terminal of the channel, and when uX,L<uX,U or uX,i<0, it moves backward and is eluted from the left terminal of the channel. If uX,L=uX,U or uX,i = 0, tR becomes infinite and the solute peak will be permanently retained within the channel. In a simplified case of ti, U = 0, Eq (6) is reduced to a familiar form, FLtR = RS = VL + KVU (7) where RS is the retention volume, and VL and VU are volume of the lower mobile phase and upper stationary phase in the column, respectively. Similarly, if ti, L = 0, Eq (6) becomes FUtR = RS = VU + VL/K (8) and the solute peak would be eluted from the left outlet of the channel. 2)Simplified Model In the simplified study, only the right half of the above separation channel is used and the sample was injected at the left terminal with the lower mobile phase for both hydrostatic and hydrodynamic systems. In order to avoid the elution of the test sample through the left terminal with the upper mobile phase, the partition coefficient was chosen to satisfy the formula FL ti, L (1+BL)/(1+KBL)>FU ti, U K(1+BU)/(1+KBU) (9). Validation of the theories Using a set of two-phase solvent systems each with a suitable test sampales, the validity of the above theories was successfully confirmed by spiral tube assembly countercurrent chromatography. Glossary AC: Cross-sectional area of the channel AL: Cross-sectional area of the channel occupied by the lower phase AL,L: Cross-sectional area of the channel occupied by the lower phase when the lower phase is mobile AL,L: Cross-sectional area of the channel occupied by the lower phase when the upper AU: Cross-sectional area of the channel occupied by the upper phase AU,U: Cross-sectional area of the channel occupied by the upper phase when the lower phase is mobile phase is mobile AU,U: Cross-sectional area of the channel occupied by the upper phase when the upper phase is mobile B: Volume ratio of upper phase to the lower phase in the channel or AU/AL BL: B for lower phase mobile or AU1/AU1;BU : B for upper phase mobile or AU2/AU2 FL: Volumetric velocity of the lower phase (ml/min) FU: Volumetric velocity of the upper phase (ml/mi) K: Partition coefficient of solute expressed by solute concentration in the upper phase divided by that of the lower phase L: Half length of the channel uL: Linear velocity of the lower phase (cm/min) uU: Linear velocity of the upper phase (cm/min) uX,L : Solute velocity (cm/min) through the channel with the lower phase is mobile uX,U : Solute velocity (cm/min) through the channel with the upper phase is mobile uX,i : Average velocity (cm/min) of solute through the channel RS: Retention volume of solute ti, L: Unit programmed time for lower phase mobile ti, U: Unit programmed time for upper phase mobile tR: Retention time of solute VC: Total column volume VL: Volume of the lower phase in the column (0.5 VC) VU: Volume of the upper phase in the column (0.5 VC)